In the Hot Springs Episode of Neon Genesis Evangelion, they use precisely this grating dialogue when they destroy an angel with the help of thermal expansion, which by amazing coincidence, Shinji had been studying that very day.
Done rather effectively in Full Metal Panic!! The Second Raid, where during a mission briefing one of the SRT members makes a joke about the Cretans Paradox. When the commanding officers realize that a mole is relaying their communications to the enemy, they manage to turn the tables by obliquely referencing the joke, which lets the field teams know to do the exact opposite of what they're ordered thereafter.
An episode of Transformers Victory had an early scene of Wingwaver and Dashtacker teaching Jean about levers. Sure enough, later in the episode, he came up with a plan to use a lever to free a trapped human.
Kakashi's lecture about revenge before the Sasuke retrieval arc of Naruto. Guess what becomes one of the central themes of the manga (and not just to Sasuke)?
During Naruto's Rasenshuriken training, he finally nails it after asking Kakashi how to "look towards right and left at the same time", whereupon Kakashi creates a shadow clone, inspiring Naruto to use shadow clones to do two things at once. The same phrase uttered word-for-word by Fukusaku later during his Sage Mode training gives him the idea to overcome his obstacle, again by using Shadow Clones.
Doesn't exactly help the protagonist solve anything, but a plot point is explained this way in The Vision of Escaflowne. Specifically, a class lecture drops off the random tidbit that Isaac Newtown studied gravity blah blah but also in a metaphysical sense. Meanwhile the main villain is revealed to be from Earth, for his true first name to be Isaac, and has created a ton of powerful inventions for Zaibach, including technobabble about the gravity of hearts and such. Hmmmmmmmmm.
A Certain Scientific Railgun: During the Level-Upper Arc, Mikoto and Kuroko discuss Synaesthesia, the phenomenon under which a certain sensory stimulus can evoke sensations in a different form, after Kuroko off-handedly comments on wind chimes making her feel cooler, or red evoking the taste of strawberries. Turns out it's the exact missing link they need in solving the Level-Upper case, and the key to defeating the AIM-Burst.
In Pokémon Special, Pearl is told by the Fan Club President that in order to prepare for Platinum's first Super Contest performance, each member of the Sinnoh Trio must do one different part. Much later on in the story, when everyone is freaking out over the revelation that Team Galactic is planning to blow up the three lakes, Pearl notices the ribbon that Platinum's Empoleon won and remembers the President's words. He then declares that each of them must go to and protect one different lake, marking the first time in their journey together that the three of them would have to travel alone.
Early in an episode of Kamichu!, Yurie is shown reviewing her social studies textbook regarding the role and mission of the Japan Self Defense Forces. Later, she uses this information to convince a group of Ground Self Defense Force troops to not interfere with the alien's escape attempt.